prove that 10n1 3 x 4n1 5 is divisible by 9 for all positi

prove that 10^(n+1) + 3 x 4^(n?1) + 5 is divisible by 9 for all positive integers n. use induction

P(n) = 10^(n+1) + 3 x 4^(n-1) + 5

Base case: P(1) = 10^(1+1) + 3 x 4^(1-1) + 5 = 10^(1+1) + 5 = 105 is divisible by 9 , which is clearly true.

Assume it\'s true for n=k; i.e., assume that 10^(k+1) + 3 x 4^(k-1) + 5 is divisible by 9 (the induction hypothesis).

for k+1, 10^(k+1+1) + 3 x 4^(k+1-1) + 5 = 10^(k+2) + 3 x 4^(k) + 5 = 100 x 10^(k) + 3 x 4^(k) + 5 is divisible by 9.